Assessment and Future Directions of Nonlinear Model Predictive ...

Sampled-Data MPC for Nonlinear Time-Varying Systems 119a “sampling-feedback” law and thus define a trajectory in a way which is verysimilar to the concept introduced in [5]. Those trajectories are, under mild conditions,well-defined even when the feedback law is discontinuous.There are in the literature a few works allowing discontinuous feedback laws inthe context of MPC. (See [8] for a survey of such works.) The essential feature ofthose frameworks to allow discontinuities is simply the sampled-data feature —appropriate use of a positive inter-sampling time, combined with an appropriateinterpretation of a solution to a discontinuous differential equation.4 Barbalat’s Lemma and VariantsBarbalat’s lemma is a well-known and powerful tool to deduce asymptotic stabilityof nonlinear systems, especially time-varying systems, using Lyapunov-likeapproaches (see e.g. [17] for a discussion and applications).Simple variants of this lemma have been used successfully to prove stability resultsfor Model Predictive Control (MPC) of nonlinear and time-varying systems[7, 15]. In fact, in all the sampled-data MPC frameworks cited above, Barbalat’slemma, or a modification of it, is used as an important step to prove stabilityof the MPC schemes. It is shown that if certain design parameters (objectivefunction, terminal set, etc.) are conveniently selected, then the value functionis monotone decreasing. Then, applying Barbalat’s lemma, attractiveness of thetrajectory of the nominal model can be established (i.e. x(t) → 0ast →∞).This stability property can be deduced for a very general class of nonlinear systems:including time-varying systems, nonholonomic systems, systems allowingdiscontinuous feedbacks, etc.A recent work on robust MPC of nonlinear systems [9] used a generalizationof Barbalat’s lemma as an important step to prove stability of the algorithm.However, it is our believe that such generalization of the lemma might provide auseful tool to analyse stability in other robust continuous-time MPC approaches,such as the one described here for time-varying systems.A standard result in Calculus states that if a function is lower bounded anddecreasing, then it converges to a limit. However, we cannot conclude whetherits derivative will decrease or not unless we impose some smoothness propertyon f(t). ˙ We have in this way a well-known form of the Barbalat’s lemma (seee.g. [17]).Lemma 1 (Barbalat’s lemma 1). Let t ↦→ F (t) be a differentiable functionwith a finite limit as t →∞.IfF˙is uniformly continuous, then F ˙ (t) → 0 ast →∞.A simple modification that has been useful in some MPC (nominal) stabilityresults [7, 15] is the following.Lemma 2 (Barbalat’s lemma 2). Let M be a continuous, positive definitefunction and x be an absolutely continuous function on IR.If‖x(·)‖ L ∞ < ∞,∫ T‖ẋ(·)‖ L ∞ < ∞, andlim T →∞ 0M(x(t)) dt < ∞, then x(t) → 0 as t →∞.